On anelliptic approximations for $qP$ velocities in transversally isotropic media

Exact expressions

Wavefront propagation in the general anisotropic media can be described with the anisotropic eikonal equation

$$v^2\left(\frac{\nabla T}{\vert\nabla T\vert},\mathbf{x}\right)\,\vert\nabla T\vert^2 = 1\;,$$ (1)

where $\mathbf{x}$ is a point in space, $T(\mathbf{x})$ is the traveltime at that point for a given source, and $v(\mathbf{n},\mathbf{x})$ is the phase velocity in the phase direction $\mathbf{n} = \frac{\nabla T}{\vert\nabla T\vert}$ .

In the case of VTI media, the three modes of elastic wave propagation ($qSH$ , $qSV$ , and $qP$ ) have the following well-known explicit expressions for the phase velocities (Gassmann, 1964):

$$v_{SH}^2(\mathbf{n},\mathbf{x}) = m\,\sin^2{\theta} + l\,\cos^2{\theta}\;;$$ (2)

$$v^2_{SV}(\mathbf{n},\mathbf{x}) = \frac{1}{2}\,\left[(a+l)\,\sin^2{\theta} + (c+l)\,\cos^2{\theta}\right] -$$

$$\frac{1}{2}\,\sqrt{\left[(a-l)\,\sin^2{\theta} - (c-l)\,\cos^2{\theta}\right]^2 + 4\,(f+l)^2\,\sin^2{\theta}\,\cos^2{\theta}}\;;$$ (3)

$$v^2_{P}(\mathbf{n},\mathbf{x}) = \frac{1}{2}\,\left[(a+l)\,\sin^2{\theta} + (c+l)\,\cos^2{\theta}\right] +$$

$$\frac{1}{2}\,\sqrt{\left[(a-l)\,\sin^2{\theta} - (c-l)\,\cos^2{\theta}\right]^2 + 4\,(f+l)^2\,\sin^2{\theta}\,\cos^2{\theta}}\;,$$ (4)

where, in the notation of Backus (1962) and Berryman (1979), $a=c_{11}$ , $c=c_{33}$ , $f=c_{13}$ , $l=c_{55}$ , $m=c_{66}$ , $c_{ij}(\mathbf{x})$ are the density-normalized components of the elastic tensor, and $\theta$ is the phase angle between the phase direction $\mathbf{n}$ and the axis of symmetry.

The group velocity describes the propagation of individual ray trajectories $\mathbf{x}(\tau)$ . It can be determined from the phase velocity using the general expression

$$\mathbf{V} = \frac{d \mathbf{x}}{d \tau} = v \mathbf{n} + \left(\mathbf{I} - \mathbf{n}\, \mathbf{n}^T\right) \nabla_{\mathbf{n}} v\;,$$ (5)

where $\mathbf{I}$ denotes the identity matrix, $\mathbf{n}^T$ stands for the transpose of $\mathbf{n}$ , and $\nabla_{\mathbf{n}} v$ is the gradient of $v$ with respect to $\mathbf{n}$ . The two terms in equation (5) are clearly orthogonal to each other. Therefore, the group velocity magnitude is (Berryman, 1979; Byun, 1984; Postma, 1955)

$$V = \vert\mathbf{V}\vert = \sqrt{v^2 + v_{\theta}^2}\;,$$ (6)

where

$$v_{\theta}^2 = \left\vert\left(\mathbf{I} - \mathbf{n}\, \mathbf{... ...right\vert^2 - \left\vert\mathbf{n} \cdot \nabla_{\mathbf{n}} v\right\vert^2\;.$$ (7)

The group velocity has a particularly simple form in the case of elliptic anisotropy. Specifically, the phase velocity squared has the quadratic form

$$v_{\mbox{ell}}^2(\mathbf{n},\mathbf{x}) = \mathbf{n}^T\,\mathbf{A}(\mathbf{x})\,\mathbf{n}$$ (8)

with a symmetric positive-definite matrix $\mathbf{A}$ , and the group velocity is

$$\mathbf{V}_{\mbox{ell}} = \mathbf{A}\,\mathbf{p}\;,$$ (9)

where $\mathbf{p} = \nabla T = \mathbf{n}/v(\mathbf{n},\mathbf{x})$ . The corresponding group slowness squared has the explicit expression

$$\frac{1}{V_{\mbox{ell}}^2(\mathbf{N},\mathbf{x})} = \mathbf{N}^T\,\mathbf{A}^{-1}(\mathbf{x})\,\mathbf{N}\;,$$ (10)

where $\mathbf{N}$ is the group direction, and $\mathbf{A}^{-1}$ is the matrix inverse of $\mathbf{A}$ . For example, the elliptic expression (2) for the phase velocity of $qSH$ waves in VTI media transforms into a completely analogous expression for the group slowness

$$\frac{1}{V_{SH}^2(\mathbf{N},\mathbf{x})} = M\,\sin^2{\Theta} + L\,\cos^2{\Theta}$$ (11)

where $M=1/m$ , $L=1/l$ , and $\Theta$ is the angle between the group direction $\mathbf{N}$ and the axis of symmetry.

The situation is more complicated in the anelliptic case. Figure 1 shows the $qP$ and $qSV$ phase velocity profiles in a transversely isotropic material - Greenhorn shale (Jones and Wang, 1981), which has the parameters $a=14.47\,$km$^2/$s$^2$ , $l=2.28\,$km$^2/$s$^2$ , $c=9.57\,$km$^2/$s$^2$ , and $f=4.51\,$km$^2/$s$^2$ . Figure 2 shows the corresponding group velocity profiles. The non-convexity of the $qSV$ phase velocity causes a multi-valued (triplicated) group velocity profile. The shapes of all the surfaces are clearly anelliptic.

exph
Figure 1. Phase velocity profiles for $qP$ (outer curve) and $qSV$ (inner curve) waves in a transversely isotropic material (Greenhorn shale).

exgr
Figure 2. Group velocity profiles for $qP$ (outer curve) and $qSV$ (inner curve) waves in a transversely isotropic material (Greenhorn shale).

A simple model of anellipticity is suggested by the Muir approximation (Dellinger et al., 1993; Muir and Dellinger, 1985), reviewed in the next section.

On anelliptic approximations for $qP$ velocities in transversally isotropic media